On Systemic Risk Levels: A Comparison Between U.S. and non-U.S. banks

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On Systemic Risk Levels: A Comparison

Between U.S. and non-U.S. banks

University of Groningen

MSc Thesis EORAS

Joram Y. van Dijk (S2926520)

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: Prof. Dr. L. Spierdijk

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On Systemic Risk Levels: A Comparison

Between U.S. and non-U.S. banks

University of Groningen

MSc Thesis EORAS

Joram Y. van Dijk (S2926520)

June 17, 2019

Abstract

In this thesis systemic risk levels in a network of large, worldwide banks will be evaluated. The aim of this thesis is to investigate the differences in systemic risk levels between the regions considered in the network. First, we obtain the ARMA-GARCH models fitting the bank stock index returns best in a marginal setting. Afterwards, we will use the standardized residuals of the marginal models as input data for a multivariate copula model. This enables us to incorporate the conditional correlations between the bank stock indices in our network. We will evaluate both multivariate copula models and R-vine copula networks. The systemic risk levels are evaluated using the conditional value-at-risk (CoVaR). To get marginal contribution levels of the banks, we use the ∆CoVaR. Lastly, a backtesting study is used to compare the performance of the multivariate copula models. We see that there are substantial differences between systemic risk levels for the different regions considered. Also, we find that U.S. banks are not the only contributors to systemic risk, but that European and U.K. banks score slightly higher. Also, European -, U.K. - and U.S. banks on average have higher systemic risk levels than Canadian - and Asian-Pacific banks. We recommend the regulator to take the regional differences into account in the determination of global systemically important banks (G-SIBs) and future regulation. We also encourage the use of vine copula models for determining systemic risk levels, but note that more research is needed to make a conclusive statement.

Keywords:

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1 Introduction

1.1 Literature Review

1.1.1 Motivation & Relevance

The financial crisis of 2007-2009, also called ”The Great Recession”, has increased the awareness of the need for active risk management for financial institutions, especially in the field of sys-temic risk. During syssys-temic financial crises, the financial system as a whole is threatened due to the tendency of losses to expand through financial institutions (Adrian and Shin, 2010). Adrian and Brunnermeier (2016) define systemic risk as ”the risk that the intermediation capacity of the entire financial system is impaired, with potentially adverse consequences for the supply of credit to the real economy”. Hence, if banks suffer large losses this could harm the role of banks in society as the supplier of credit due to expansion of losses through financial institutions (Adrian and Shin, 2010). A first component of systemic risk is the presence of negative risk spillover effects, or financial contagion. This is the spreading of distress by so-called individually sys-temic institutions due to their size and interconnectedness (Adrian and Brunnermeier, 2016). Second, there are institutions that are not necessarily Systemically Important Financial Insti-tutions (SIFI), but are systemic as part of a herd (Adrian and Brunnermeier, 2016).

Systemic risk assessment, monitoring and regulation is of key importance for Central Banks due to their coordinating role (Freixas et al., 2000). If a bank becomes insolvent and has to close, this may have negative systemic consequences for other banks. As Freixas et al. (2000) noted, the orderly closure of insolvent banks is the responsibility of the Central Bank. Thorough re-search on the dependencies between financial institutions during a crisis has been conducted in recent years. In general, this revealed that conditional correlations of financial institutions tend to be stronger in distress and also that the aforementioned spillover effects do occur (Longin and Solnik, 2001; Adrian and Shin, 2010). Because of these tight connections and large dependen-cies between banks, it is crucial to restrict the impact of systemic crises, evoking in more banks becoming insolvent. Therefore, the Central Bank must provide its liquidity to the counterparts of the insolvent bank to limit spillover effects (Freixas et al., 2000).

Initially, the Basel I and II regulation focussed mainly on the risk of financial institutions in isolation. However, to protect the financial system as a whole, regulation should also focus on the stability of the financial system. In light of the 2007-2009 financial crisis, the Basel III regu-lation also tries to restrain systemic risk by introducing some new reguregu-lation (King and Tarbert, 2011). In this new regulation, the FSB and BCBS decide on a list of global systemically impor-tant banks (G-SIBs) with additional capital requirements (Hull, 2018).1,2 The most important indicator of a G-SIB is its size, as they are considered ”too big to fail”. Therefore, they have to be bailed out in case of extreme distress (Hull, 2018). Current regulation by the BCBS already includes the interconnectedness, complexity, substitutability and cross-jurisdictional activity as other determinants of systemic risk levels, as there was critique on basing this method solely on the size of a bank (Zhou, 2010; Glasserman et al., 2015). Using these determinants, a bank-specific score is obtained. Based on this score, banks are placed in a certain bucket, with an implied additional capital requirement. Besides the regulation of the BCBS, national regulators also give extra capital requirements to banks that have not been identified as G-SIB, but can be seen as domestic systemically important banks (D-SIBs). Despite these efforts, Glasserman et al. (2015) find that banks with higher scores not necessarily have higher capital buffers relative to banks with lower scores. This stresses the difficulty and necessity in determining systemic risk levels (more specifically, the differences in systemic risk levels between banks both before,

1

Financial Stability Board (FSB) and the Basel Committee on Banking Supervision (BCBS).

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during and after the financial crisis). Hence, the regulator could benefit from more insights in the determination and height of systemic risk levels.

Furthermore, Hull (2018) notes that countries are subject to the same Basel III regulation. However, local regulators have some discretion in using the rules and can implement its own supervisory system. Also, even more importantly, the different legislation across all countries aggravates the regulatory issues even more. Generally, differences with Basel III regulation do occur in local regulation. For example, in the U.S. the Dodd-Frank Act is of great regulatory importance and shows some differences with Basel III regulation (Hull, 2018). Where European regulators usually comply with the Basel regulation, U.S. regulators are generally somewhat more reluctant in this. This is aggravated by the complicated process involved in implementing regulation. The different structure of specific banks complicates this matter even further. To make additional regulation successful, good measures of systemic risk levels are needed.

A decade ago, the failure of the U.S. bank Lehman Brothers in 2008 had an enormous impact on the financial system, due to its systemic importance (Fernando et al., 2012). Also, we see that the list of G-SIBs is currently being dominated by U.S. banks. Therefore, it would be informative to see, using systemic risk measures, if U.S. banks are nowadays systemically more important compared to banks from other countries. If this is the case, additional regulation could help restrict the impact of U.S. bank spillover effects caused by their systemic impor-tance. Therefore, it would be informative to see what the differences in systemic risk levels are between geographical regions. Such new measures, and consequently regulation, could help tackle the problem of systemic risk at its source: the main global systematically important banks. In sum, systemic risk can cause problematic loss spillover effects, especially in periods of severe distress. Therefore, it is of key importance to determine G-SIBs, so that additional regulation is successful in restricting systemic risk. Especially the regulator could benefit from these insights. However, the determination of systemically important banks and differences in systemic risk levels is not flawless nowadays. The regulatory differences between countries complicate this matter even further. Therefore, there is still work to be done in this area.

1.1.2 Modelling approaches

In order to assess the level of systemic risk, the joint return distribution of the systems bank stock index returns has to be modelled. By doing so, conditional correlations can be taken into account. In a first modelling stage, the marginal return distributions have to be modelled (Brechmann and Czado, 2013). Afterwards, the conditional correlations have to be accounted for in a model combining the marginal distributions (Brechmann and Czado, 2013; Dißmann et al., 2013). This two-step approach has been performed by Brechmann and Czado (2013). As an alternative to these stages, a multivariate distribution can be used. For both methods a risk measure is needed to acquire the desired systemic risk levels from the model.

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structure (Bollerslev et al., 1988; Engle, 2002), the exploding number of parameters to estimate is often an issue.

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1.2 Research Question & Strategy

The main goal of this thesis is answering the following research question: ”What is the dif-ference in systemic risk levels between U.S. banks and non-U.S. banks during and after the Great Recession?” With ”the Great Recession”, we refer to the financial crisis of 2007-2009. To answer the research question, we need to decide on the modelling of the stock index returns. Therefore, we will consider the following subquestions:

• On the basis of the AIC, what ARMA-GARCH model should be used to model the marginal return distributions of our bank stock index returns?

• On the basis of backtesting statistics, which copula model performs best in modelling sys-temic risk using the CoV aR in our network of banks?

In this thesis, we will assess the level of systemic risk in a network containing the largest 25 con-stituents of the MSCI World Banks Index in the period 2006-2018, containing several important, worldwide regions. We will use the two-step approach as performed by Brechmann and Czado (2013). The ARMA-GARCH model will be used as the marginal distribution of stock index returns, with the number of lags chosen according to the Akaike information criterion (AIC). Looking at our evaluation criteria and using quasi-maximum likelihood, we will investigate what error distribution is best for the ARMA-GARCH model. We will model the conditional corre-lations, based on the standardized residuals from the marginal models, with the R-vine copula model and compare the results to other benchmark multivariate copula models. In this stage, it is important that most of the dependencies are taken into account. The value of systemic risk will be measured by the CoVaR and the ∆CoVaR and for contrast purposes be compared to the VaR. The performance of the copula models will be evaluated using backtesting statistics for the VaR and CoVaR, in a similar fashion as Girardi and Erg¨un (2013) and Brechmann and Czado (2013). We will look at the proportion of failure (POF) test statistic as introduced by Kupiec (1995), the duration-based Weibull test for independence test as introduced by Christoffersen and Pelletier (2004) and a conditional coverage test combining the previous two test principles, as introduced by Christoffersen (1998).

1.2.1 Hypotheses

We expect that systemic risk levels based on the CoVaR in general are higher before the financial crisis than during and just after the financial crisis, as Adrian and Brunnermeier (2016) found that systemic risk typically increases before a crisis and emerges only when a crisis occurs. Also, we expect U.S. banks to have relative high systemic risk levels compared to banks from other countries (Glasserman et al., 2015). Several authors have found good performance of vine copula models (Brechmann and Czado, 2013; Dißmann et al., 2013)). However, we remain cautious in drawing conclusions, as vine copula models are still in their beginning stage and there is also critique on vine copula models (Acar et al., 2012).

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2 Definitions

In this section the definitions and theory needed to understand the methodology are presented. This section will be purely theoretical. Section 3 will build on this section by explaining our methodology for the analysis of the data.

2.1 Marginal model

As discussed in section 1.2, we will use the ARMA-GARCH model for the marginal distributions of the returns. Using the notation of McNeil et al. (2015), the process X = {Xt; t = 1, . . . , T }

is an ARMA(p1, q1)-GARCH(p2, q2) model if it satisfies equation (1)-(3):

Xt= µt+ σtzt, (1) µt= µ + p1 X i=1 φi(Xt−i− µ) + q1 X j=1 θj(Xt−j− µt−j), (2) σ_t2 = ω + p2 X i=1 αi(Xt−i− µt−i)2+ q2 X j=1 βjσt−j2 , (3)

for (p1, q1, p2, q2) the number of lags included in the ARMA-GARCH model, (µt, µ, φi, θj)0∈ R4

∀ t = 1, . . . , T , i = 1, . . . , p₁ and j = 1, . . . , q1. Also, we require ω > 0, αi ≥ 0, βj ≥ 0 for

i = 1, . . . , p2 and j = 1, . . . , q2. Here, µt is the ARMA-process for the mean and σ2t is the

GARCH-process for the volatility. We define the residual process (also referred to as innovation process) Z = {zt, t = 1, . . . , T } to be a strict white noise process with a mean equal to zero and

a variance equal to 1, i.e. Z ∼ SWN(0, 1). A strict white noise (SWN) process is defined as a series of iid (independent and identically distributed) random variables with a finite variance. The ARMA(p1, q1)-GARCH(p2, q2) model is covariance stationary if Pi=1p2 αi+Pqj=12 βj < 1.

For the innovation process Z a distribution is used, for example a Gaussian distribution or a Student’s t distribution.

In some cases, it is better to use the ARMA-eGARCH model (Nelson, 1991) as marginal distri-bution of the returns. This enables modelling asymmetric effects if necessary. Equation (1) and (2) are still used for the ARMA-eGARCH model, but equation (3) is replaced by equation (4):

log(σ_t2) = ω +

p2

X

i=1

(αizt−j+ γi(|zt−i| − E|zt−i|)) + q2

X

j=1

βjlog(σ2t−j), (4)

where γj ∈ R covers the size effect, αi covers the sign effect and zt−i, for t ∈ {0, 1, . . . , T }, are

realizations of the innovation process Z.

2.2 Multivariate model

In this section we will start with the basic requirements of the multivariate model. The R-vine copula model is more complex than the multivariate copula models. This model will be further defined in section 2.2.1. In sections 3.2.1 and 3.2.2 the definitions are used to explain how we create our multivariate copula model and our R-vine copula model.

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Definition 2.1 (Copula) A d-dimensional copula is a distribution function C : [0, 1]d_{→ [0, 1].}

The following three properties must hold: (1) C(u1, . . . , ud) = 0 if ui = 0 for any i.

(2) C(1, . . . , 1, ui, 1, . . . , 1) = ui for all i ∈ {1, . . . , d}, ui∈ [0, 1].

(3) The rectangle inequality should hold (see McNeil et al. (2015) for more details).

Hence, a d-dimensional copula is a multivariate cumulative distribution function (CDF) on [0, 1]d with standard uniform marginal distributions. Note that for the R-vine copula model we only use bivariate copulae (i.e. we set d = 2). Modelling univariate distributions and the multivariate distribution separately is exactly what our two-step method is based on. The theorem by Sklar (1959), given in theorem 2.1, focusses on this elegance:

Theorem 2.1 (Sklar) Let F be a joint distribution function with margins F1, . . . , Fd. Then

there exists a copula C : [0, 1]d→ [0, 1] such that, for all x1, . . . , xd in ¯R = [−∞, ∞],

F (x1, . . . , xd) = C(F1(x1), . . . , Fd(xd)). (5)

Conversely, if C is a copula and F1, . . . , Fd are univariate distribution functions, then the

func-tion F defined in equafunc-tion (5) is a joint distribufunc-tion funcfunc-tion with margins F1, . . . , Fd.

Using theorem 2.1, we can write

C(u1, . . . , ud) = F (F1−1(u1), F −1

2 (u2), . . . , F −1

n (ud)), (6)

with F_i−1 the inverse of the marginal distribution function Fi for i = 1, . . . , d. In section 3.2 we

will use theorem 2.1 to explain how to use the results of the marginal model in the multivariate model.

To evaluate the dependencies we have several available measures. The most common dependence measure is Pearson’s Correlation, as defined in equation (7):

ρ(X, Y ) = Cov(X, Y )

pVar(X)pVar(Y ), (7)

for X, Y two random variables with finite variance. However, Pearson’s Correlation is only a measure of linear dependence and it is not invariant under non-linear, strictly increasing transformations (Hendrich, 2012). Therefore, it is good to instead look at rank-based measures of association. These measures also take into account non-linear dependence. We will use Kendall’s τ in our evaluation of dependencies. It basically is the probability of concordance minus the probability of discordance. Kendall’s τ correlation measure ρτ is defined in definition

2.2:

Definition 2.2 (Kendall’s tau) Let (X1, X2) be a random vector and ( ˜X1, ˜X2) a second vector

with the same distribution, independent of the former vector. Kendall’s τ is defined as

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2.2.1 R-vine copula model

According to Joe and Kurowicka (2011), a vine is a graphical tool that is used for labeling constraints in high-dimensional distributions. A special case of a vine, introduced by Bedford and Cooke (2002), is a regular vine (R-vine), in which all constraints are two-dimensional or conditional two-dimensional. A definition of a Regular vine is given by Dißmann et al. (2013): Definition 2.3 (Regular vine) V = (T1, . . . , Tn−1) is a Regular vine on n elements if

(1) T1 is a tree with nodes N1 = {1, . . . , n} and an accompanying set of edges E1.

(2) For i = 2, . . . , n − 1, Ti is a tree with nodes Ni = Ei−1 and an edge set Ei.

(3) For i = 2, . . . , n − 1 and {a, b} ∈ Ei with a = {a1, a2} and b = {b1, b2} it must hold that

#(a ∩ b) = 1.

The last characteristic is called the proximity condition, ensuring that two nodes in tree k + 1 can only be tied by an edge if they have a common node in tree k. Hence, a regular vine on n variables consists of n − 1 trees, in which the edges in tree k are the nodes of tree k + 1. Two special cases of regular vines are the canonical vine (C-vine) and D-vine. According to Kurowicka and Cooke (2006), in a C-vine each tree has one node with edges to all other nodes in the network. In a D-vine each node in the first tree has at most two edges to other nodes. However, as C-vines and D-vines are more restrictive than R-vines, we will focus on R-vines. Later, we will give an example of a R-vine copula model.

First, we will combine this network with copulae. For this, Aas et al. (2009) defined a pair-copula decomposition. This is a way of extending bivariate pair-copulae to a multivariate model. By doing so, they circumvent the problem of the scarcity of higher-dimensional copulae.

Using the notation of Aas et al. (2009), for n random variables X1, . . . , Xn with joint density

f (x1, . . . , xn) we can factorize the density as

f (x1, . . . , xn) = f (xn) · f (xn−1, xn) f (xn) ·f (xn−2, xn−1, xn) f (xn−1, xn) · · · f (x1, x2, . . . , xn) f (x2, . . . , xn) (9) = f (xn) · f (xn−1|xn) · f (xn−2|xn−1, xn) · · · f (x1|x2, . . . , xn). (10)

Now, we follow Aas et al. (2009): using equation (6), for f the joint density function and for F absolutely continuous with strictly increasing and continuous F1, . . . , Fn, we get

f (x1, . . . , xn) = c12···n(F1(x1), . . . , Fn(xn)) · f1(x1) · · · fn(xn), (11)

where c12···n is a n-dimensional copula density. Following the reasoning of Aas et al. (2009), for

c12(·, ·), c12|3(·, ·) and c13|2(·, ·) some appropriate pair-copula densities, it follows

f (x1|x2) = c12(F1(x1), F2(x2)) · f1(x1), (12)

f (x1|x2, x3) = c12|3(F1|3(x1|x3), F2|3(x2|x3)) · f (x1|x3), (13)

f (x1|x2, x3) = c13|2(F1|2(x1|x2), F3|2(x3|x2)) · f (x1|x2). (14)

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decomposed into a term consisting of a pair-copula and a conditional marginal density with the formulas in equation (15) and (16) given by Bedford and Cooke (2001):

. (16)

where v is a n-dimensional vector, vj is the j-th element of v, v−j is the vector v without its j-th

element and C_xv_j|v−j is a bivariate copula distribution function. We define cje,ke|De as a bivariate

copula density, e as an edge from the set of edges Eifor i = 1, . . . , n−1, Deas the conditioning set

and je, keas the conditioned variables. Note that the conditioned variables, jeand ke, are always

two variables, whereas De always contains i − 1 elements for tree Ti−1. We use xDe to denote

the vector of elements De from x. Combining all previous results, Kurowicka and Cooke (2006)

find that the R-vine density f (x1, . . . , xn) and R-vine copula density c(F1(x1), . . . , Fn(xn)) for

trees T1, . . . , Tn−1 are given by equation (17) and (18):

To make R-vine copula models more intuitive, we provide an illustration in example 2.1. For more details on pair-copula constructions Kurowicka and Cooke (2006), Aas et al. (2009) and Dißmann et al. (2013) can be consulted.

Example 2.1 Here we give an example of a R-vine copula model. In the example we use 6 variables, xi for i = 1, . . . , 6. The R-vine trees are given in figure 1. The corresponding R-vine

matrix is given by the matrix in equation (19):

        4 0 0 0 0 0 6 6 0 0 0 0 5 5 5 0 0 0 1 1 1 1 0 0 2 2 3 3 3 0 3 3 2 2 2 2         . (19)

Each element of the matrix corresponds to a specific pair-copula in the joint density. As we have seen, for each copula in the joint density we will have two elements as the conditioned variables. The conditioned set contains i − 1 elements for tree i, i = 1, 2, 3, 4, 5. First consider element (4,2) (fourth row, second column) of the R-vine matrix. This element is variable 1 and will be in the conditioned set. The other element in the conditioned set is the diagonal element in the same column, hence variable 6. Now, the conditioning set consists of all elements below element (4,2). Hence, variables 2 and 3 are in the conditioning set for this pair-copula. Then, the pair-copula corresponding to element (4,2) is c16|23(F1|23(x1|x2, x3), F6|23(x6|x2, x3)). We see

that this copula does also occur as an edge in the third tree in figure 1. The other elements of the matrix can be explained in the same way, and match with the other edges in the R-vine trees in figure 1.

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f123456(x1, . . . , x6) =f1(x1) · f2(x2) · f3(x3) · f4(x4) · f5(x5) · f6(x6) ·c12(F1(x1), F2(x2)) · c23(F2(x2), F3(x3)) · c34(F3(x3), F4(x4)) ·c₂₅(F2(x2), F5(x5)) · c36(F3(x3), F6(x6)) ·c_13|2(F1|2(x1|x2), F3|2(x3|x2)) · c24|3(F2|3(x2|x3), F4|3(x4|x3)) ·c_35|2(F3|2(x3|x2), F5|2(x5|x2)) · c26|3(F2|3(x2|x3), F6|3(x6|x3)) ·c_14|23(F_1|23(x1|x2, x3), F4|23(x4|x2, x3)) · c16|23(F1|23(x1|x2, x3), F6|23(x6|x2, x3)) ·c_15|23(F1|23(x1|x2, x3), F5|23(x5|x2, x3)) ·c_45|123(F_4|123(x4|x1, x2, x3), F5|123(x5|x1, x2, x3)) ·c_56|123(F5|123(x5|x1, x2, x3), F6|123(x6|x1, x2, x3)) ·c_46|1235(F_4|1235(x4|x1, x2, x3, x5), F6|1235(x6|x1, x2, x3, x5)). 1 2 3 4 5 6 1,2 2,3 2,5 3,6 3,4 1,2 2,3 3,4 2,5 3,6 1, 3|2 3, 5|2 2, 6|3 2, 4|3 2, 4|3 1, 3|2 2, 6|3 3, 5|2 1, 4|23 1, 5|23 1, 6|23 1, 4|23 4, 5|123 1, 5|23 5, 6|123 1, 6|23 4, 5|123 4, 6|1235 5, 6|123

Figure 1: R-vine trees for example 2.1. The left panel represents trees 1-3 respectively, the right panel represents trees 4 and 5.

2.3 Systemic risk measure

The VaR is defined as the α-quantile of the loss distribution, i.e. VaRi_α,t= F_i−1(α), where F_i−1 is the generalized inverse of the loss distribution, i denotes the specific bank and t the date. Equation (20) gives the relationship that holds for the VaR (McNeil et al., 2015):

P (Li,t ≤ VaRiα,t) = α. (20)

The CoVaR uses the definition of the VaR. There are two different definitions in the literature. The original definition by Adrian and Brunnermeier (2016) is given in equation (21). The alternative definition by Girardi and Erg¨un (2013) is given in equation (22).

P (Ls,t≤ CoVaRs|iα,tα,A|Li,t = VaRiα,t) = α, (21)

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Hence, CoVaRs|iα

α,t is the VaR of the financial system, s (MSCI World Banks index), conditional

on some event for VaRi_α,t of bank i at confidence level α, indicated by iα, at time t. Hence, iα

indicates an event for bank i at VaRi_α,t. For Adrian and Brunnermeier (2016) this event is the return of bank i being at VaRi_α,t, whereas for Girardi and Erg¨un (2013) this event is the loss of bank i being at most at VaRi_α,t. As Girardi and Erg¨un (2013) note, there are several benefits for using this alternative definition. Firstly, it allows for losses further in the tail. Secondly, it enables us to backtest the CoVaR estimates in the same way as VaR estimates. For more information and appealing properties of the CoVaR Adrian and Brunnermeier (2016) can be consulted.

To translate the CoVaR risk measure in a contribution by a single institution, we will use the ∆CoVaR as introduced by Adrian and Brunnermeier (2016) and redefined by Girardi and Erg¨un (2013). It is defined in equation (23):

∆CoVaRs|i_α,t = 100 × (CoVaRs|iα

α,t − CoVaR s|i0.5

α,t )/CoVaR s|i0.5

α,t . (23)

Hence, ∆CoVaRs,i_α,t is the difference between the VaR of the system, s, conditional on a stress situation for bank i and the VaR of the system conditional on bank j being at its median state, VaRi_0.5, at time t. Therefore, the ∆CoVaRs,i_α,t measures the marginal contribution of a single institution to the overall systemic risk level.

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3 Methodology

In this section, the methodology needed to perform our analysis is presented. We will build on section 2, where the basic theoretical necessities are presented. We will take these characteristics into account in choosing the best method to perform the analysis. First, the method for obtaining the marginal distributions is given. Afterwards, the methods used to model the conditional correlations using the results from the marginal distributions will be presented. Lastly, the method for obtaining the level of systemic risk will be given. The theory and definitions needed to understand the methodology have been discussed in section 2.

3.1 Marginal model

We will use the ARMA-GARCH model, as defined in section 2.1, for the marginal distribu-tions. It is not the goal of this thesis to present an exploratory analysis of the wide range of ARMA-GARCH models. Therefore, we will slightly restrict the choices we make regarding the modelling of the marginals. For the GARCH part, we will set the number of lags at p2, q2= 1.

For the ARMA part, we will only look at lags p1, q1 ∈ {0, 1, 2, 3}. Doing so, we ensure that the

number of lags will not explode and therefore our model remains parsimonious. Also, too many lags results in an increasing probability of overfitting. Later, we will shortly evaluate if more lags have a large impact on the AIC, to see if this restriction harms the outcome of our subquestion given in section 1.2. However, in previous research these restrictions have also been used on financial data with satisfactory results (Hendrich, 2012; Girardi and Erg¨un, 2013; Brechmann and Czado, 2013). We will not make any assumptions for the innovation process Z, as we will use quasi-maximum likelihood in the fitting process (McNeil et al., 2015). For more information on the possible innovation models McNeil et al. (2015) can be consulted. Also, if a Sign Bias test (Engle and Ng, 1993) gives a significant outcome, this indicates that there might be asymmetric effects in the data. To model these asymmetric effects we will evaluate the performance of the ARMA-eGARCH model, as defined in section 2.1.

Our selection of the ARMA-(e)GARCH model for the marginal return distributions can be summarized by the following steps, based on Hendrich (2012) and Brechmann and Czado (2013): • We fit ARMA(p1, q1)-GARCH(1,1) models for p1, q1 ∈ {0, 1, 2, 3} for each return series

using quasi-maximum likelihood (McNeil et al., 2015). Doing so, we do not need to make any assumptions about the innovation distribution, ensuring more robust results (White, 1982).

• The performance of the distributions is evaluated based on the AIC (Akaike, 1973). Several other studies also used the AIC as a performance measure (Hendrich, 2012; Reboredo and Ugolini, 2015b). We will choose the distribution with the lowest AIC value. We will also look at the BIC and log-likelihood (LLH) values, where we prefer the minimal and maximal value respectively.

• We look at the QQ-plot given by a specific distribution, and only proceed with the distri-bution if the theoretical quantiles and the sample quantiles are similar (i.e. if the points forming the QQ-plot lie on a straight line). If the differences are too big, we will look for a better distribution. We will also look at a plot of the empirical densities of the standardized residuals.

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an ARCH-LM test (Engle, 1982) to see if there is any remaining autocorrelation in the squared residuals. Lastly, we perform several Sign Bias tests (Engle and Ng, 1993) to see if our model needs to account for asymmetric effects. If so, we will use the eGARCH model as defined in equation (4).

If we have performed all steps mentioned above, we can move on to the multivariate model. In the multivariate model we will use the standardized residuals of the marginal distributions as input data.

Now that we have explained the method for obtaining the right marginal distributions, we need a model to evaluate the dependencies in our network of banks. As mentioned before we will evaluate dependencies for the standardized residuals, not directly for the returns. To make the standardized residuals suitable for evaluation with a copula model, we need to ensure they are uniformly distributed on [0,1]. To do so, we will perform a probability integral transform (Brechmann et al., 2013; McNeil et al., 2015). More specifically, we denote by zi,t the

standard-ized residual for bank i ∈ {1,2,. . . ,26} at time t = 1, . . . , n, where n is our last data point on 12-28-2018. In matrix notation, we get the n × 26 dimensional matrix ˆU as copula input data. The resulting copula input data ˆuj,t can be observed in equation (24):

ˆ

ui,t= ˆFi(zi,t), for i ∈ {1,2,. . . ,26} , t = 1, . . . , n, (24)

where ˆFi denotes cumulative distribution function of the innovation process in the specific

ARMA-GARCH model for bank i. By applying equation (24) we ensure that our copula data is uniformly distributed, because cumulative distribution functions satisfy F : R → [0, 1] (McNeil et al., 2015). By separating the univariate distributions and multivariate distribution we are able to estimate the marginal distributions and dependencies separately.

Applying theorem 2.1, we use the input data ˆui,t for i ∈ {1, 2, . . . , 26} and t = 1, . . . , n to find a

multivariate copula distribution C that fits the dependency patterns in the residuals best. We will separate the remaining part of this section in a part covering the multivariate copula model and the R-vine copula model.

3.2.1 Multivariate copula model

First, we explain how to use multivariate copula models for evaluating the dependence structure in our bank network. We already discussed how to obtain the required data for the copula model in the previous section. We will continue with this data.

As Brechmann and Czado (2013) note, there is only a limited number of parametric multivariate copula families with flexible dependence. Therefore, and due to the large size of our bank net-work, we expect that multivariate copula models will be outperformed by R-vine copula models. Nevertheless, we will also discuss multivariate copula models, to have a benchmark model to evaluate the performance of the R-vine copula model.

Our network of banks consists of 25 bank stock indices and the MSCI World Banks index, resulting in a dimension of 26 for our multivariate copula model. We use theorem 2.1 to obtain the multivariate copula model. Here C denotes our chosen copula distribution, Fi denotes the

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will be transformed to ui using equation (24), for all i = 1, . . . , 26). Following Brechmann and

Czado (2013), we find that the joint density is given by equation (25):

f (x1, . . . , x26) = "₂₆ Y k=1 fk(xk) # · c(F1(x1), . . . , F26(x26)). (25)

We will focus on some elliptical copulae and Archimedean copulae. For more information on these copula models please consult Joe and Kurowicka (2011) or McNeil et al. (2015). The different multivariate copulae we evaluate are the Gaussian -, Student’s t -, Clayton -, Frank -, Joe - and Gumbel copula. Most multivariate copula models are inflexible regarding dependency modelling. The Gaussian copula is incapable of modelling tail dependence, the Student’s t cop-ula is not able to model tail asymmetry and the Archimedean copcop-ulae also have their own lacks (Brechmann and Czado, 2013; McNeil et al., 2015). We expect to encounter these shortcomings in the results. We will use maximum likelihood estimation to find the desired copula parameters.

3.2.2 R-vine copula model

Vine copula models are a way to model the dependencies in our bank network. The basics of (R-)vine copula models have been explained in section 2.2.1. Here we will explain how we will fit a R-vine copula model as a multivariate model for our observations.

Because of the bivariate nature of R-vine networks, they are a flexible tool for high-dimensional dependence modelling. Hence, they are very useful to model the dependencies in our network of banks. We will build on the definitions given in sections 2.2 and 2.2.1 to explain how we use the R-vine copula model in our bank network. In a R-vine copula model we use bivariate copulae to model the conditional correlations. Thus, we can choose a specific bivariate copula to model the conditional correlation between two banks, instead of using the same multivariate copula for all conditional correlations. This makes vine copula models a flexible choice to model dependencies (Brechmann et al., 2013).

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Copula Family N t C F G J BB1 BB7 RC RG

Positive dependence -

-Negative dependence - - - -

-Tail asymmetry - -

-Lower tail dependence - - -

-Upper tail dependence - - - -

-Table 1: This table contains the most important bivariate copulae considered in the modelling of the dependence structure of our network of banks, with their dependence characteristics as given by Brechmann and Czado (2013). The copulae are: N = Gaussian, t = Student’s t, C = Clayton, F = Frank, G = Gumbel, J = Joe, BB1 = Clayton-Gumbel, BB7 = Joe-Clayton, RC = rotated Clayton (90◦), RG = rotated Gumbel (90◦).

The possibilities with R-vine copula models are numerous. First of all, we have to choose whether we want to use the same bivariate copula for all conditional correlations. Considering the fact that all copulae have their own drawbacks in this case (see table 1), we choose to use a mixed R-vine copula model. Thus, we use different copulae for different conditional correlations. However, as Morales-N´apoles et al. (2010) note, the number of possible R-vine copula models is enormous. If we have to fit a bivariate copula model for every edge in our trees, we get a lot of different copulae and parameters. Therefore we need some simplification of the model to keep it tractable.

A first step can be taken by using the independence copula for small dependencies. An inde-pendence test based on the empirical Kendall’s τ as proposed by Genest and Favre (2007) is performed during the estimation. If the independence test is not rejected, the independence copula will be used for the pair of banks rather than some other copula. Doing so, several authors (Brechmann et al., 2012; Dißmann et al., 2013) find that the structure of the model can be extensively simplified at a low cost regarding the evaluation criteria (AIC, log-likelihood). Therefore, we will include the option to use the independence copula in our model.

Another possible simplification is the trunctation of R-vines as introduced by Brechmann et al. (2012). This means that for the remaining trees, starting from a certain tree K, all pair-copulas are replaced by the independence copula. This makes sense, as we expect to capture most depen-dencies in the first trees (Hendrich, 2012). Hence, in example 2.1 truncation after level 3 would mean that for trees 4 and 5 we replace copulae C45|123, C56|123 and C46|1235for the independence

copula. The difficulty here lies in determining the truncation level K. Brechmann et al. (2012) use a sequential procedure based on the Vuong test by Vuong (1989), possibly with AIC/BIC correction, to determine the suitable truncation level. They find promising results for a R-vine copula model, but Hendrich (2012) find less promising results. We will test for truncation, but remain cautious in drawing conclusions and simplifying the model too much.

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Hendrich (2012) or Dißmann et al. (2013) can be consulted. After the sequential method full maximum likelihood will be applied to obtain the final R-vine copula model (Dißmann et al., 2013).

3.3 Systemic risk measure

Now that we have modelled the conditional correlations, we need a measure of systemic risk to evaluate the differences between the banks in our network. As mentioned in section 1, previous research revealed that the VaR is an improper way to evaluate systemic risk levels (Adrian and Brunnermeier, 2016). Therefore, we will use the CoVaR, originally introduced by Adrian and Brunnermeier (2016). The required definition is given in section 2.3. In this thesis, we will use the definition given by Girardi and Erg¨un (2013) in equation (22) because of the appealing properties mentioned in section 2.3. Therefore, if we use the CoVaR, we refer to the CoVaR from equation (22).

We will obtain the CoVaR by first sampling from our R-vine copula and multivariate copula models. Equation (1) - (4) will be used to obtain returns. Afterwards, we will use equation (20), (22) and (23) to get the desired measures. For more details on this method please consult section A in the appendix and Aas et al. (2009); Hendrich (2012); Dißmann et al. (2013).

3.4 Model evaluation

For a first comparison between the different R-vine copula and multivariate copula models we will use the AIC, BIC and log-likelihood. In previous research (Brechmann, 2010; Hendrich, 2012; Dißmann et al., 2013) a popular way to compare different vine copula models was to use the Vuong test (Vuong, 1989), which compares two different models directly. Therefore, we will also consider this test. Note that the Vuong test is also used to obtain the truncation level K in a R-vine copula model.

Besides using the aforementioned evaluation criteria, it is also important to see how the VaR and CoVaR perform in a backtesting study. Therefore, we will first choose the best R-vine copula model and the best multivariate copula model based on the aforementioned evaluation criteria, and afterwards compare them in a backtesting study. We will consider the POF test (Kupiec, 1995), the duration-based Weibull test for independence (Christoffersen and Pelletier, 2004) and a joint conditional coverage test combining the previous two test principles (Christof-fersen, 1998). Regarding the independence test, a popular test is the Markov independence test as introduced by Christoffersen (1998). However, several authors found that the duration-based Weibull test (Christoffersen and Pelletier, 2004) performed better. We will follow the method-ology of Brechmann and Czado (2013) in the backtesting study. More details can be found in section A in the appendix.

3.5 Other methods

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matrix is positive definite. Adrian and Brunnermeier (2016) used quantile regressions to obtain the CoVaR and found promising results.

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4 Data

In this section, the data used in this thesis will be described. Also, the manipulations necessary to make the data suitable for our analysis will be shortly mentioned. Afterwards, to give a good impression of our dataset, we will visualize some aspects of our dataset.

4.1 Data description

We collect data for daily adjusted closing prices in the period January 2006 up to January 2019 for the MSCI World Banks Index and the 25 largest constituents of the MSCI World Banks Index, resulting in thirteen years of daily observations. The MSCI World Banks Index consists of 90 large worldwide banks from 23 developed market countries.3 Weights corresponding to the 90 banks are reviewed quarterly. The initial dataset was obtained from the Thomas Reuters Eikon database. Furthermore, the headquarters locations of our banks can be observed in figure 7 in the appendix. Moreover, the banks corresponding to the abbreviations used throughout this thesis, together with the weight of each bank (on April 4th2019) in the MSCI World Banks index can be consulted in table 13 in the appendix. We see that a large part of the MSCI World Banks index is constituted of U.S. banks.

For some banks we have a few missing observations in the examined period. For each bank we removed the observations corresponding to the dates on which at least one bank misses an observation. By doing so, we will get the same number of observations for all banks and obser-vations for the same dates for all banks. Afterwards, the adjusted closing prices are converted to daily returns using equation (26):

Ri,t= log(Pi,t+1) − log(Pi,t), (26)

where Ri,t denotes the return in period t of bank i and Pi,t denotes the closing price of bank i in

period t. In total, this results in n = 2920 observed returns for 25 different bank stock indices and the MSCI World Banks Index. The losses are obtained as the negative of the returns. In table 2 some characteristics of our dataset can be observed. We observe that, for most banks, the percentage of extreme losses is below 1%. Also, we see that for most banks the percentage of losses is around 50%. Note that we are looking at the statistics of losses here: a negative value at the mean and minimum indicates a return, whereas a positive value indicates a loss. We see that the skewness is close to zero for most banks, meaning that we would not have to take skewness into account in the modelling stage. However, for some banks (PNC FINL Services Group (PNC), Lloyds Banking Group (LLOYDS)) we see that the skewness is larger. This is mainly driven by the large losses these banks have experienced (53.44% and 78.68% respectively). For the kurtosis, we expect a value of 3 for the normal distribution. We see that for all banks we have a larger kurtosis, indicating that the distribution of the returns for these banks will be more peaked and have fat tails. As a consequence, the normal distribution is expected to have a poor fit, as it lacks these characteristics. What further impact this will have on the marginal distribution of the bank stock index returns, we will see in the following sections.

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Bank Summary Statistics

Mean (%) Sd (%) Min (%) Max (%) Skewness Kurtosis Loss (%) Ext. (%) MSCI 0.02 1.60 -12.87 14.10 0.08 12.20 48.05 0.07 JPM -0.03 2.57 -27.03 23.23 -0.42 17.72 48.84 0.45 BAC 0.02 3.51 -41.40 34.21 -0.18 26.59 48.73 1.03 WFC -0.01 2.77 -28.34 27.21 -0.86 23.80 49.62 0.68 HSBC 0.01 1.82 -14.42 20.80 0.34 15.97 50.21 0.10 C 0.08 3.72 -47.79 49.47 0.16 35.27 50.07 1.23 RBOC -0.02 1.74 -14.42 16.36 0.12 9.94 47.50 0.10 TDB -0.03 1.40 -12.34 13.63 -0.13 13.15 46.92 0.07 CWBA -0.02 1.55 -11.90 14.05 0.37 8.18 47.53 0.07 USB -0.01 2.24 -20.57 20.05 0.18 16.89 47.60 0.48 SAN 0.02 2.40 -20.88 22.17 0.31 9.55 47.98 0.38 NOVA -0.01 1.46 -12.18 14.31 -0.08 12.58 47.50 0.07 WEST -0.00 1.67 -11.89 11.82 0.15 5.01 48.05 0.03 MITS 0.04 2.34 -14.26 15.83 -0.25 4.41 49.66 0.10 PNC -0.02 2.67 -31.55 53.44 1.68 69.02 48.97 0.48 BNP 0.02 2.71 -18.98 30.86 0.33 13.57 49.08 0.55 ANZ -0.00 1.73 -17.63 11.57 -0.16 9.50 47.91 0.10 LLOYDS 0.05 3.55 -40.79 78.68 3.63 101.25 50.45 1.03 NAB 0.01 1.76 -16.00 14.49 0.20 9.26 48.36 0.10 SUM 0.04 2.44 -15.57 16.53 -0.26 6.84 49.28 0.27 MONT -0.01 1.48 -11.79 13.08 -0.51 12.80 46.23 0.03 ING 0.03 3.25 -27.66 32.14 0.04 15.34 49.18 0.99 MIZU 0.06 2.38 -16.74 16.03 -0.23 7.69 48.15 0.31 BBT -0.00 2.42 -21.20 26.61 0.15 17.09 48.49 0.34 BBVA 0.03 2.34 -19.91 17.65 0.03 7.64 50.31 0.21 CIB -0.01 1.56 -13.15 13.29 -0.35 11.46 47.88 0.03

Table 2: This table contains summary statistics of the losses in our dataset. Besides some basic characteristics, also the skewness, kurtosis, the percentage of losses and the percentage of extreme losses are included. We define an extreme loss to be a loss of more than 10%. Note that this characteristic is only used for the explorative data analysis.

4.2 Data visualization

Before we start working on our analysis, it is good to first visualize some important character-istics of our dataset. First, we included QQ-plots based on the normal distribution in figure 2. We already found evidence of the poor fit of the normal distribution for our data in table 2. Here we again see that our data has fatter tails and is more leptokurtic.

To see if our data shows volatility clustering, we included the loss series in figure 3. Note that we already transformed the data, meaning that upward peaks have to be associated with losses. We can see that, for all banks, the volatility in the losses is higher during the financial crisis, as we would expect. Also, we see massive volatility clustering: extreme observations occur in groups. Especially during the crisis we see that there are large ups and downs in the observations. We note that the scaling in the different graphs differs. Therefore, it seems that banks like Lloyds Banking Group (LLOYDS) and PNC FINL Services Group (PNC) are less volatile. However, this is due to the different scaling and their outliers. We can conclude that there is massive volatility clustering in the losses for all banks.

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Lastly, we included empirical Kendall’s τ values in table 3. The sequential method that is per-formed to create the R-vine network, as discussed in section 3.2.2, uses the empirical Kendall’s τ . Therefore, it is useful to have a first insight in the values of the empirical Kendall’s τ as a dependence measure. The banks are ordered based on their geographical region to better visualize dependence patterns within and between regions. So, the first five banks (on the y-axis) are Canadian banks, the next seven banks are U.S. banks, the next seven banks are from Europe and the U.K. (with the MSCI World Banks index placed in the middle) and the last seven banks are from the Asian-Pacific group (which is again separated in Asian and Pacific banks). In the table, darker colours are associated with a higher Kendall’s τ value, and thus a larger dependency between the banks. On the other hand, brighter colours tell us that the dependency between the banks is weaker.

The Kendall’s τ values vary between 0.06 and 0.68, and thus are all positive. We can see that the dependencies are strongest within the geographical groups. For all groups, the dependencies are larger within the group than between groups, as was expected. What also is interesting to see, is that between the Asia and Pacific group the dependencies are in general slightly higher than be-tween the other regions. This same pattern can be observed for the U.S. -, Canadian -, European - and U.K. group. Especially between the U.S. group and the Canadian group we can see that the values are in general slightly higher than the values between the U.S. group and other groups. It is also interesting to see what the Kendall’s τ values are for the MSCI World Banks index. We can see that the values are largest for the European banks BBVA, Banco Santander (SAN), BNP Paribas (BNP) and ING Groep (ING). The values for the U.S. and U.K. are slightly lower, but still above 0.40. The values for the Canadian banks are around 0.35 and for the Asian-Pacific group around 0.25. Hence, based on this first, basic analysis, it seems that European banks have the largest dependency with the MSCI World Banks index. Also, we can conclude that the geographical region plays a large role, both within the groups and between the groups.

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Table 3: This table contains empirical Kendall’s τ correlation values in a heat map. The banks are ordered in groups based on their geographical region, to better visualize the dependence patterns within and between regions. The MSCI World Banks index is placed in the Europe group.

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5 Empirical Results

In this section, the results obtained by conducting the methodology mentioned above will be presented. The results of the marginal distributions and the results of the multivariate models, to incorporate the conditional correlations in our model, are discussed. Afterwards, the sys-temic risk levels for the different banks will be discussed, and their validity will be tested in a backtesting study.

5.1 Marginal model

We follow the steps as given in section 3.1. The results of the marginal models are given in table 4. The best performing model is specified in the first columns. Furthermore, the number of lags in the ARMA part are given by p1 and p2. Column ’e’ indicates whether the eGARCH model

was used rather than the standard GARCH model and ’Distribution’ indicates the resulting innovation process. Moreover, the parameters are given in the next columns. The parameters are tested using a t-test. For this test, H0 states that a parameter is equal to zero, whereas

Ha states that a parameter is significantly different from zero. Based on this test, a significant

outcome on a 95% confidence level (meaning that we find a p-value < 0.05) is indicated by a bold value, meaning that we reject H0. Note that we use robust standard errors in the t-test.

The parameters ˆµ, ˆφ and ˆθ are used in the ARMA part, as defined in equation (2), whereas the parameters ˆω, ˆα and ˆβ are used in the GARCH-part, as defined in equation (3). The parameter ˆ

γ is used only in the eGARCH model, as defined in equation (4), ˆλ is used only for the GHYP distribution, ˆζ is the skewness parameter for the GHYP and NIG distributions and ˆν is the shape parameter.

Looking at the results, we observe that all models have at most one lag included for the au-toregressive - (AR) and moving-average (MA) part. Sometimes more lags did slightly decrease the AIC. However, we want to have a parsimonious model. Therefore, we will not replace a parsimonious model for a slightly lower AIC. Also, we observe that the eGARCH model was used for most banks. The performance of the eGARCH model was only evaluated if the Sign Bias tests indicated that the marginal model should take asymmetric effects into account. Also, we see that the Student’s t (STD) distribution was used in some cases, but that the Normal Inverse Gaussian (NIG) - and Generalized Hyperbolic (GHYP) distribution are superior. As expected, for none of the banks the Gaussian distribution was used. This again signals that financial return data require more leptokurtic distributions.

Furthermore, we see that for the majority of the models ˆα and ˆβ are significant. Except for ˆ

µ, the other parameters are in most of the cases significant. The shape parameter ˆν is only insignificant for the models where a GHYP distribution was chosen. The skewness parameter

ˆ

ζ is in the range [−0.136, 0.941] and not always significantly different from zero. Hence, not all models require a skewed innovation distribution. Nevertheless, several models perform signif-icantly better by allowing for skewness. The asymmetry parameter ˆγ is only insignificant for US Bancorp (USB). Regarding the stationarity of the models, we require that ˆα + ˆβ < 1 for the standard GARCH model, and we require ˆβ < 1 for the eGARCH model (Hendrich, 2012; McNeil et al., 2015). We see that in all cases the chosen model is stationary.

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Besides the evaluation criteria in table 4, we also looked at the QQ-plots and plots of the empir-ical densities of the standardized residuals of the marginal models. The plots can be observed in figure 8 in the appendix. For the QQ-plots, we see that most points lie on the straight line or are close to the straight line, indicating good performance of the distribution used. For Banco Santander (SAN), Lloyds Banking Group (LLOYDS) and Bank Montreal (MONT) we find that in the far right tail the theoretical quantile does not match the sample quantile perfectly. How-ever, in the evaluation process for these banks, it turned out that changes in the marginal model did not improve the performance in the QQ-plot. Therefore, we do not change the specified distributions. Regarding the empirical densities, we clearly see that the distributions that were chosen (yellow) have a much better fit than the normal distribution (blue), as we already ob-served in section 4.2.

The last step in the performance evaluation is to check what the performed tests tell us about the chosen distribution. The results of the performed tests can be observed in table 5. We per-formed a Ljung-Box test with lag 20 on the standardized residuals and the squared standardized residuals. Formally, H0 states that the (squared) standardized residuals of the fitted model are

independently distributed, whereas Ha states that the (squared) standardized residuals of the

fitted model show serial dependence. Therefore, we do not want to reject H0, as a rejection

would indicate that there is still autocorrelation in our (squared) standardized residuals. For a 95% confidence level this means that we prefer p-values larger than 0.05. All bold values in table 5 indicate a significant outcome (p-value < 0.05). We see a significant outcome for LB20 only for Sumitomo Mitsui Financial Group (SUM), and a significant outcome for LB220

only for Commonwealth Bank of Australia (CWBA), Mitsubishi Financial Group (MITS), PNC FINL Services Group (PNC) and Canadian Imperial Bank (CIB). As alternative models did not (or not without harming some other important characteristic of the model) improve the test outcomes, we choose to accept this drawback. Also, the plots in figure 8 do not show any big problems.

Also, several Sign Bias tests are used to see if an eGARCH model could improve the perfor-mance. In general, H0 states that there are no asymmetries in the returns and Ha states that

there are asymmetries in the returns, with slight differences regarding the form of asymmetry between the tests. If the Sign Bias tests gives a significant outcome based on a 95% confidence level (meaning that we reject H0), we try the eGARCH model. In some cases the Sign Bias

tests of the best performing model still gives a significant outcome on a 95% confidence level. However, for these banks we did already improve the outcome of the test by using the eGARCH model, or using the eGARCH model did not significantly improve the performance of the model. Lastly, we performed ARCH-LM tests for lags two, five and ten. Formally, H0 states that the

model removes ARCH effects in the returns present before fitting the model and Ha states

that after the fitting process there still are ARCH effects in the squared residuals of the model. Therefore, we want the ARCH-LM test to give an insignificant outcome (meaning that we fail to reject H0), as a significant outcome would mean that there are still ARCH effects in the

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Bank Model Parameters Evaluation criteria

p1 p2 e Distribution µˆ φˆ θˆ ωˆ αˆ βˆ γˆ ζˆ νˆ λˆ LLH AIC BIC

MSCI 0 1 NIG 0.000 - 0.128 -0.090 0.084 0.990 0.132 0.124 2.087 - 8943.854 -6.120 -6.104 JPM 0 1 - NIG -0.001 - -0.055 0.000 0.091 0.904 - -0.005 1.175 - 7754.800 -5.307 -5.292 BAC 0 0 NIG 0.000 - - -0.063 0.064 0.992 0.186 0.000 1.087 - 7289.079 -4.988 -4.973 WFC 0 0 - NIG 0.000 - - 0.000 0.094 0.905 - -0.034 1.109 - 7888.621 -5.400 -5.387 HSBC 0 1 STD 0.000 - 0.035 -0.093 0.046 0.989 0.152 - 4.937 - 8359.946 -5.721 -5.707 C 1 1 STD 0.000 -0.482 0.495 -0.071 0.068 0.991 0.200 - 4.737 - 7263.837 -4.970 -4.953 RBOC 0 0 GHYP 0.000 - - -0.068 0.075 0.992 0.110 0.514 0.686 -3.175 8548.847 -5.850 -5.834 TDB 0 1 - GHYP -0.001 - 0.023 0.000 0.095 0.902 - 0.438 0.254 -2.656 9280.592 -6.351 -6.335 CWBA 1 1 GHYP 0.000 0.913 -0.926 -0.086 0.037 0.990 0.147 0.915 0.250 -3.913 8677.418 -5.937 -5.916 USB 0 1 NIG 0.000 - -0.051 -0.063 0.077 0.993 0.154 0.107 1.574 - 8394.809 -5.744 -5.728 SAN 0 0 - NIG 0.000 - - 0.000 0.079 0.910 - 0.094 1.533 - 7331.355 -5.017 -5.005 NOVA 0 1 - NIG 0.000 - 0.053 0.000 0.084 0.912 - 0.152 1.867 - 9144.225 -6.258 -6.244 WEST 0 1 GHYP 0.000 - 0.047 -0.078 0.049 0.991 0.138 0.941 0.250 -4.008 8355.174 -5.717 -5.698 MITS 0 1 NIG 0.000 - 0.088 -0.137 0.054 0.982 0.167 -0.136 2.755 - 7273.787 -4.977 -4.960 PNC 1 1 NIG 0.000 0.669 -0.724 -0.057 0.048 0.993 0.147 0.051 1.163 - 7874.159 -5.387 -5.369 BNP 0 1 STD 0.000 - 0.039 -0.061 0.087 0.992 0.109 - 7.041 - 7190.261 -4.920 -4.906 ANZ 1 0 GHYP 0.000 0.062 - -0.058 0.052 0.993 0.117 0.842 0.250 -3.099 8384.125 -5.736 -5.718 LLOYDS 1 1 - STD 0.000 -0.513 0.544 0.000 0.085 0.914 - - 5.249 - 7262.388 -4.969 -4.955 NAB 1 0 GHYP 0.000 0.050 - -0.108 0.048 0.987 0.175 0.884 0.250 -3.246 8375.923 -5.731 -5.712 SUM 1 0 NIG 0.000 0.075 - -0.106 0.055 0.986 0.166 -0.112 2.101 - 7338.148 -5.021 -5.004 MONT 1 1 NIG 0.000 0.347 -0.279 -0.083 0.066 0.991 0.148 0.148 1.120 - 9335.969 -6.388 -6.370 ING 0 1 STD 0.000 - 0.035 -0.067 0.095 0.991 0.152 - 6.413 - 7060.222 -4.831 -4.817 MIZU 1 1 - STD 0.000 0.205 -0.135 0.000 0.082 0.917 - - 5.269 - 7575.453 -5.184 -5.170 BBT 0 1 - GHYP 0.000 - -0.055 0.000 0.061 0.938 - 0.469 0.479 -2.524 7908.956 -5.412 -5.395 BBVA 0 1 - STD 0.000 - 0.050 0.000 0.068 0.921 - - 5.927 - 7340.275 -5.023 -5.011 CIB 0 1 NIG 0.000 - 0.047 -0.043 0.052 0.995 0.110 0.089 1.234 - 9168.303 -6.274 -6.258

Table 4: This table contains the outcomes of the fitting process of the marginal models. Note that all models have a GARCH(1,1) as a basis and the ARMA lags p1 and

p2 can take on values in the set {0, 1, 2, 3}. Also, ’e’ indicates whether the eGARCH model was used. The resulting distribution types fall in the group of the Student’s t

- (STD), Normal Inverse Gaussian - (NIG) and Generalized Hyperbolic (GHYP) distribution. Furthermore, the estimated parameters and the evaluation criteria are given. Significance of the parameters, based on the t-test as described in this section, is indicated by a bold value.

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Bank Ljung-Box Sign Bias ARCH-LM LB20 LB220 SB NSB PSB JE 2 5 10 MSCI 0.435 0.530 0.151 0.338 0.234 0.440 0.173 0.308 0.351 JPM 0.771 0.131 0.212 0.512 0.101 0.012 0.222 0.246 0.207 BAC 0.386 0.190 0.150 0.605 0.629 0.333 0.260 0.366 0.500 WFC 0.186 0.789 0.581 0.464 0.223 0.204 0.858 0.856 0.953 HSBC 0.868 0.739 0.117 0.219 0.520 0.427 0.731 0.844 0.812 C 0.992 0.195 0.252 0.519 0.111 0.109 0.943 0.930 0.653 RBOC 0.394 0.325 0.636 0.287 0.527 0.635 0.503 0.135 0.144 TDB 0.354 0.940 0.578 0.589 0.320 0.266 0.517 0.582 0.747 CWBA 0.087 0.001 0.190 0.125 0.432 0.208 0.299 0.035 0.059 USB 0.126 0.369 0.361 0.928 0.894 0.742 0.820 0.756 0.887 SAN 0.328 0.737 0.761 0.771 0.241 0.686 0.614 0.592 0.664 NOVA 0.423 0.961 0.468 0.364 0.330 0.112 0.872 0.919 0.918 WEST 0.785 0.239 0.210 0.454 0.160 0.456 0.540 0.704 0.540 MITS 0.636 0.015 0.053 0.689 0.121 0.105 0.772 0.081 0.060 PNC 0.230 0.013 0.353 0.929 0.000 0.005 0.708 0.649 0.798 BNP 0.103 0.997 0.398 0.793 0.276 0.654 0.648 0.959 0.992 ANZ 0.858 0.149 0.826 0.478 0.061 0.233 0.627 0.890 0.976 LLOYDS 0.111 0.997 0.349 0.694 0.621 0.435 0.672 0.942 0.976 NAB 0.556 0.340 0.235 0.560 0.162 0.079 0.763 0.562 0.280 SUM 0.046 0.587 0.174 0.867 0.206 0.491 0.536 0.496 0.636 MONT 0.700 0.622 0.354 0.805 0.324 0.208 0.995 0.809 0.622 ING 0.781 0.218 0.018 0.633 0.081 0.094 0.039 0.099 0.105 MIZU 0.967 0.628 0.552 0.509 0.355 0.218 0.920 0.941 0.870 BBT 0.745 0.359 0.200 0.435 0.935 0.540 0.900 0.886 0.964 BBVA 0.570 0.067 0.948 0.588 0.088 0.129 0.330 0.349 0.382 CIB 0.905 0.000 0.959 0.093 0.257 0.248 0.193 0.440 0.207

Table 5: This table contains the results of the performed tests on the fitted marginal models. Ljung-Box tests are performed on the standardized residuals and squared standardized residuals with lags equal to twenty. The Sign Bias tests are performed on the standardized residuals (SB = Sign Bias, NSB = Negative Sign Bias, PSB = Positive Sign Bias and JE = Joint Effect.). The ARCH-LM test is performed on the standardized residuals for two, five and ten lags. A bold value indicates a significant outcome on a 95% confidence level.

Now that we have determined the marginal distribution functions that we will use for the banks in our network (see table 4), we can investigate the performance of the multivariate models. As mentioned in section 3.2, we will evaluate the performance of both multivariate copula mod-els and R-vine copula modmod-els. First, we transform the standardized residuals zi,t for bank

i ∈ {1, 2, . . . , 26} at time t = 1, . . . , n into the copula input data ˆui,t, as explained in section 3.2.

This data is then used in the copula models to get to the results.

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the Gaussian copula model for all evaluation criteria. This is what was expected, as the Gaus-sian copula can’t account for tail dependence (Brechmann and Czado, 2013; McNeil et al., 2015). For the R-vine copula models we consider several models. As mentioned in section 3.2.2, R-vine copula models can become very complicated and difficult to interpret. Therefore, we introduced two simplification methods: truncation of the model after a certain tree and the use of an independence test. The latter will be evaluated in the results in table 6, the former will be evaluated in table 7. In table 14 in the appendix all pair-copula types used in the fitting process can be consulted. Note that the models in table 6 are without truncation. We considered R-vine pair-copula models with:

• all possible bivariate copula types from table 14 (R-vine);

• only unrotated copulae 0-10 from table 14 (R-vine Large subset); • only copulae 0-6 from table 14 (R-vine Small subset);

• all bivariate copulae from table 14, without an independence test (R-vine No Indepen-dence);

• only Gaussian copulae (R-vine Gaussian); • only Student’s t copulae (R-vine Student’s t);

• only Gaussian and Student’s t copulae (R-vine Gaus. & Std.).

Note that only the R-vine No Independence model does not use an independence test. In the lower part of table 6 the results of the vine copula models are presented. We see that the R-vine model without independence test performs best regarding the AIC, BIC and log-likelihood. However, the increase in parameters compared to the R-vine model with independence test is dramatically (35%). We see that the subsets considered are not far off from the full R-vine model based on the evaluation criteria. Especially for the simple R-vine model with only Gaussian -, Student’s t - and independence copula this is remarkable.

To evaluate the R-vine models on more than just the AIC, BIC and log-likelihood, we also performed Vuong tests. The test results are presented in the last two columns of table 6. Ba-sically, H0 states that the two tested models are similar, whereas Ha states that one model

is significantly better in explaining the characteristics in the data. A bold value indicates a significant test at the 95% confidence level, meaning that we reject H0. Therefore, one model is

significantly better than the other. Also, a positive, significant test statistic indicates that the test is in favour of the full R-vine model with an independence test. We see that, based on this test, most R-vine models are inferior to the full R-vine model with independence test. The full R-vine model without an independence test has a negative test statistic for the uncorrected test, but the test statistic for the corrected test is positive. We choose to trust the corrected test, as it corrects for possible bias. Also, the test for the R-vine model with the large subset of copulae is not significant. This indicates that neither the full R-vine model, nor the R-vine model with the large subset of copulae is preferred. Then, based on the other criteria, we choose the full R-vine model to be superior. Brechmann (2010), Hendrich (2012) and Dißmann et al. (2013) find similar results.

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can conclude that, at first sight, the R-vine copula model performs very well compared to these benchmark multivariate models. In section 5.4 we will see if this is also the case in a backtesting study for the systemic risk measures.

Furthermore, we considered truncation of the model after a certain tree as possible simplification method. In table 7 the results are given. The first column gives the last tree that was included in the R-vine model. The results of the uncorrected and Schwarz bias corrected Vuong tests are given in the next columns. Note that the full R-vine model with independence test is the benchmark model here. Therefore, a positive significant test statistic means we prefer the full R-vine model. We can observe that, for the uncorrected tests, the R-vine models truncated after level 18 or higher do not differ significantly from the full R-vine model. For the Schwarz bias corrected test, we see that the R-vine models truncated after level 13 or higher do not differ significantly from the full R-vine model. However, it should be noted that the truncated models are not superior to the full R-vine model. For truncation levels below 13, the full model is superior based on both tests.

As the full R-vine model without truncation and with independence test allowed performs best in the class of R-vine copula models, we will further evaluate this model. In figure 5 the first tree of the R-vine model is presented. The edge labels represent Kendall’s τ values as deducted from the pair-copula parameters. We see that, as expected, the MSCI is the center of the tree. Furthermore, we observe again that the geographical region has a massive impact on the struc-ture. We see that the European -, Japanese -, U.S. -, Pacific - and Canadian banks grouped in a different branch of the tree. This might indicate that distress of a bank will mostly affect banks in the same region, and banks from other global regions to a lesser extent. Also, this might be the translated effect of banks in different regions being subject to other regulation. Nevertheless, we observed in table 3 that correlations between e.g. U.S. -, European -, Canadian - and U.K. banks are not negligible. Therefore, if the distress of a bank will be transferred to other banks in the same region, in the end, it is very likely that it is also transferred to banks from other regions. This can result in spillover effects and systemic crises, as was indicated by Longin and Solnik (2001) and Adrian and Shin (2010). However, the correlations are not strong enough to be translated to the R-vine network in figure 5. Moreover, as mentioned before, Longin and Solnik (2001) found that conditional correlations tend to be stronger in financial distress. However, we can’t conclude anything about the differences in the correlations between crisis -and non-crisis years from the network in figure 5. We also see that U.K. banks Lloyds Banking Group (LLOYDS) and HSBC form single branches, and are not part of a group. This indicates that the regional pattern is less visible for U.K. banks. Nevertheless, this does not mean that U.K. banks can’t be systemically important. Regarding the Kendall’s τ values presented, we observe the same patterns as in table 3. We see that the European branch has the strongest connection with the MSCI World Banks system index, and that Asian-Pacific branches score considerably lower.

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Copula model AIC BIC LLH No. of Vuong test param. Uncor. Schwarz

Gaussian -68,440 -66,497 34,545 325 - -Student’s t -71,970 -70,021 36,311 326 - -Gumbel -22,920 -22,914 11,461 1 - -Clayton -25,559 -25,553 12,780 1 - -Frank -25,331 -25,325 12,667 1 - -Joe -21,416 -21,410 10,709 1 - -R-vine -73,718 -72,044 37,139 280 -

-R-vine Large subset -73,678 -72,046 37,112 273 1.6 -0.1

R-vine Small subset -73,647 -72,038 37,092 269 2.5 0.2

R-vine No Independence -73,961 -71,438 37,403 422 -9.1 10.5

R-vine Gaussian -67,551 -66,511 33,950 174 18.8 16.3

R-vine Student’s t -73,382 -71,456 37,013 322 2.7 6.2

R-vine Gaus. & Std. -73,602 -71,820 37,099 298 0.9 2.5

Table 6: This table contains the evaluation criteria of the multivariate copula models. The log-likelihood (LLH), AIC, BIC and number of parameters (No. of param.) are given. The last columns are Vuong tests for the R-vine copula models, uncorrected (Uncor.) and bias corrected (Schwarz). A bold value indicates significance at a 95% confidence level.

Trunc. Vuong uncor. Vuong Schwarz

Level Statistic P-value Statistic P-value

10 11.70 0.00 3.92 0.00 11 9.85 0.00 3.04 0.00 12 8.06 0.00 2.05 0.04 13 6.50 0.00 0.91 0.37 14 5.75 0.00 0.65 0.51 15 5.26 0.00 0.87 0.38 16 4.04 0.00 0.10 0.92 17 3.49 0.00 0.12 0.90 18 1.91 0.06 -0.22 0.83 19 1.49 0.14 0.15 0.88 20 1.49 0.14 0.15 0.88 21 1.49 0.14 0.15 0.88 22 0.00 1.00 0.00 1.00 23 0.00 1.00 0.00 1.00 24 0.00 1.00 0.00 1.00 25 0.00 1.00 0.00 1.00 26 0.00 1.00 0.00 1.00

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Figure 5: This is a graph of the first R-vine tree corresponding to the full R-vine model without truncation and with an independence test. The ticker names and corresponding banks are given in table 13 in the appendix. The edge labels are the Kendall’s τ values as deducted from the pair-copula parameters.

Tree Π N t C G F J Rest 1 1 0 18 0 0 6 0 1 2 2 0 18 0 0 1 0 5 3 5 0 15 0 0 3 0 3 4 10 0 9 0 0 5 0 2 5 11 1 5 0 0 3 0 6 6 18 1 1 0 0 5 0 1 7 15 1 3 2 0 3 0 2 8 19 1 3 0 0 1 0 2 9 19 3 1 0 0 1 0 2 10 19 0 2 0 0 3 0 2 11-26 384 1 8 0 0 12 0 11 Total 503 8 83 2 0 43 0 37

On Systemic Risk Levels: A Comparison Between U.S. and non-U.S. banks

On Systemic Risk Levels: A Comparison

Between U.S. and non-U.S. banks

University of Groningen

On Systemic Risk Levels: A Comparison

Between U.S. and non-U.S. banks

University of Groningen

Contents

1

Introduction

2

Definitions

3

Methodology

4

Data

5

Empirical Results